In previous posts, I had used the difference between the multi-model mean and observations in the 20th century to try and determine a simple noise model, and then extend this to see what kind of confidence intervals we could expect for the 21st century A1B projections. The result of this AR(1) process used to simulate noise was pretty successful for the 20th century, in that it gave tighter bounds around the MMM but still seemed to keep the observations inside the 54 runs. Extending it to the include 2001-2010 projection suggested that observations were on the lower end for the 2001-2010 period, but the major three datasets were within the 2.5% – 97.5 % confidence interval.

However, one of the things that the models do a poor job of simulating are ENSO variations. This can have an effect on a multi-year period, and certainly affects annual anomalies. So, I was curious about removing the ENSO component from the variations and seeing how that affected the error in the hindcast. In fact, this was part of the reason why I recently ran some regressions at different lag times with various other factors.

For this, I took a slightly different approach, which was to assume that the bulk of the error between the “forced component”/MMM and actual observations was simply the result of ENSO (since the model hindcasts include solar and volcanic forcings) and regressed the error against the Nino 3.4 index. Because the MMM does not properly simulate at sub-annual time scales I had to aggregate to the annual level after lagging Nino3.4 by a few months. This method is cheating a bit because the way this is set up means that removing a Nino3.4 component could never increase the error. However, it would probably not be accurate to regress against the entire 20th century and assume an underlying linear trend, particularly because we’d need to include solar, man-made aerosol, and volcanic forcings as well to determine what ENSO has contributed. Besides, this is more of a what-if situation where we’re wondering what extra noise would look like IF the MMM represents the “forced component” and IF the remainder of the error is primarily the result of ENSO.

Anyhow, the script is available here.

After regressing the 20th century annual errors against Nino3.4, the best fit I get is .082 * Nino3.4 (lagged 5 months):

The variance of the error between the unadjusted HadCRUT obs and the MMM is about .0150, compared to .0118 when we adjust HadCRUT for ENSO based on our fit. Not a giant improvement, and so it is doubtful that ENSO itself would explain the errors in the models around the 1940-1960 period.

But from about 1975 on the errors shrink quite a bit, improved significantly by the Nino fit:

During this time period, the best fit is .089 * Nino3.4 (lagged 6 months), so fairly similar to our entire period, and a similar coefficient/magnitude to what we were getting for Nino in the regressions solely against observations in the previous post. Here the variance in errors goes from .00769 to .00261 when we perform the ENSO adjustment, a pretty large improvement even considering the lower DF. Running AR on the residuals suggests that the noise here is more likely to be white than from an AR(1) process, and the small variance in the residuals further suggests we can get yet tighter bounds during this more “accurate” 1975-1999 period.

The distribution of these remaining errors don’t look quite normal, but I’ll use the approximation anyway for my MC sims:

Similar to what I did previously, I will create new “runs” of the models based on the MMM + noise for bootstrapping the confidence interval. First, a look at 1975-2010 if I use this white noise model with the above variance:

As can be seen, the yellow “model runs” are much tighter around the MMM than both the AR(1) noise model runs did and the actual CMIP3 individual runs. However, what is interesting is that while the black line representing the observations often jumps outside these bounds in the 1975-1999 period, the green ENSO-adjusted observations stay much closer within it. It is only in recent years that they have left the area.

So, what do our trends look like from 2000-2010 if we use this white noise model?

The resulting confidence interval for 2.5% – 97% is [0.112, 0.290] C/decade, and the Nino3.4 adjusted trend for HadCRUTv3 is well outside of it at .0037.

Of course, we have fitted Nino3.4 to the 1975-2000 errors, so it’s no surprise that we get a smaller error there than we would expect for the 21st century projections, which have not been used for training. Furthermore, the models don’t have solar and volcanic forcings for the forecast the way they did for the hindcast, so that might be another source of divergence. Then there is the question of whether actual forcings have tracked the A1B scenario, and of course there are other temperature data sets (I may try this with GISS).

Still, what’s interesting to me is how small of a noise model would explain the errors between the MMM and the ENSO-adjusted HadCRUT observations from 1975-2000. It seems particularly surprising given the way the projections and actual observations diverge early in the 21st century.